%%This set of notes needs more inter-note linking. There are lots of unlinked mentions here.%% # Index [[Analytic continuation]] [[Cauchy principal value]] [[Cauchy's integral formula]] [[Cauchy's theorem]] [[Closed contour integral]] [[Complex functions]] [[conformal map]] [[Contour integrals]] [[Dirichlet integral]] [[Holomorphic functions]] [[Jordan's lemma]] [[Laurent series]] [[Liouville's theorem]] [[Meromorphic functions]] [[Pole]] [[Real integral on the complex plane]] [[Regular functions]] [[Residue theorem]] [[Residues]] [[Riemann surfaces]] [single valued function](single%20valued%20function) [[Singularity]] [U(1)](U(1).md) --- # Proofs and examples [[Examples of contour integrals]] [Jordan's lemma](Jordan's%20lemma.md) --- # Basic concepts ## The complex plane %%Here you need to either quote an entry on the complex number field or define the complex number field here. something idk.%% ## Numbers on a complex plane A complex number $z$ is given in terms of [the complex plane](Complex%20analysis%20(index).md#The%20Complex%20plane) coordinates as $z = x + iy$ or in polar coordinates as $z = |z|e^{i\theta} = Re^{i\theta}$ where $R$ is the radius from the origin. ## Functions with complex arguments  ### Complex arguments The angle $\theta$ is a real number referred to as the _argument_ where it may be denoted as $\theta = \mathrm{Arg}(z) = \tan^{-1}\bigg(\frac{y}{x}\bigg)$ And due to the periodicity of $e^{i\theta}$ (meaning that $\mathrm{Arg}$ isn't single valued) we more formally define $\mathrm{Arg}$ as $\Theta = \mathrm{Arg}(z)$ where $\Theta = \theta + 2\pi n$ for ($n\in \mathbb{Z}$). ### Complex exponentials ### Complex logarithms The [natural log](Analysis%20(index)#Natural%20log) containing a [complex argument](Complex%20analysis%20(index).md#Complex%20arguments) is as follows. $\ln{z} = \ln{r}+i\theta = \ln{r}+i\theta = \ln{r}+i(\Theta+2\pi n)$ where its _principle branch_ is given as $\mathrm{Ln}(z)=\mathrm{r} + i\Theta \;\;\;\;\;\; (r>0, -\pi < \Theta < \pi)$ ### Derivatives of complex functions ### Complex infinite sums ### Trigonometric functions with complex arguments #### Hyperbolic trigonometric functions with complex arguments ## Euler's formula Euler's formula is given as $e^{ix}=\cos{x}+i\sin{x}$^abb760 Euler's formula provides a way of decomposing a [complex exponential](Complex%20analysis%20(index).md#Complex%20exponentials) into its [real and complex components.](Complex%20analysis%20(index).md#Numbers%20on%20a%20complex%20plane) ### Euler's identity --- # Recommended reading The basic algebraic, geometric, and analytic properties of [complex numbers of functions](Complex%20analysis%20(index).md#Basic%20Concepts) are introduced in * [Brown, J. W., Churchill R. V., _Complex Variables and Applications_, McGraw Hill, 8th edition, 2009.](Brown,%20J.%20W.,%20Churchill%20R.%20V.,%20Complex%20Variables%20and%20Applications,%20McGraw%20Hill,%208th%20edition,%202009..md), Chapters 1 and 2. Chapter 1. introduces the basics of algebraic manipulation, graphing, and the representation of complex numbers and vectors as exponentials. Chapter 2. discusses functions of complex numbers in its first half followed by limits and derivatives of complex numbers. This text is aimed at students studying mathematics with no prior knowledge of complex numbers. Thus the treatment of the subject matter is proof-based and the remainder of the text is focused on introducing the basics of complex analysis as a whole. Each section within the chapters are short and followed by exercises. A comprehensive introduction to complex numbers and functions with complex numbers is given in: * [Boas M., _Mathematical Methods in the Physical Sciences_. John Wiley and Sons, 3rd edition, 2006.](Boas%20M.,%20Mathematical%20Methods%20in%20the%20Physical%20Sciences.%20John%20Wiley%20and%20Sons,%203rd%20edition,%202006..md) Chapter 2. This chapter includes a fair number of exercises on representing complex numbers as exponentials as well as on the complex plane. In addition there are sections on Infinite series with complex numbers, trigonometry and hyperbolic trigonometry with complex arguments, as well as Euler's formula and its associated properties. * Chapter 14 of Boas M., _Mathematical Methods in the Physical Sciences_ introduces Analysis with complex numbers and formalizes the notion complex arguments in addition to an introducing complex analysis itself. --- # Bibliography [Altland, A. von Delft, J. _Mathematics for Physicists_, Cambridge University Press, 2019](Altland,%20A.%20von%20Delft,%20J.%20Mathematics%20for%20Physicists,%20Cambridge%20University%20Press,%202019.md) [Boas M., _Mathematical Methods in the Physical Sciences_. John Wiley and Sons, 3rd edition, 2006.](Boas%20M.,%20Mathematical%20Methods%20in%20the%20Physical%20Sciences.%20John%20Wiley%20and%20Sons,%203rd%20edition,%202006..md) [Brown, J. W., Churchill R. V., _Complex Variables and Applications_, McGraw Hill, 8th edition, 2009.](Brown,%20J.%20W.,%20Churchill%20R.%20V.,%20Complex%20Variables%20and%20Applications,%20McGraw%20Hill,%208th%20edition,%202009..md) Murayama H., MH2801: Complex Methods for the Sciences  Orloff, J., Lecture Notes, MIT 18.04 _Complex Variables with Applications_  Arken G. B., _Mathematical methods for Physicists_ %20-%20libgen.lc.pdf) #MathematicalFoundations/Analysis/ComplexAnalysis #MathematicalFoundations/Analysis/ComplexAnalysis/Functions #MathematicalFoundations/Analysis/ComplexAnalysis/Integrals #Bibliography #index